Morphogenesis of a Toroidal Skywalk
Riassunto
Questo lavoro di tesi si sviluppa intorno all’ideazione, sviluppo e ottimizzazione di un camminamento nel vuoto altresì chiamato come Skywalk. Il progetto consiste in una passerella pedonale dal guscio toroidale con spirali avvolgenti in acciaio che collaborano con i pannelli in vetro nel portare i carichi agenti. La struttura è stata modella su Grasshopper, un plug-in di Rhino3D che permette di generare delle geometria in maniera parametrica. Questo ambiente di programmazione ha permesso anche l’interfacciamento di add-ons, sia proprietari che creati ad hoc dall’autore. Con questi strumenti, l’autore è riuscito a creare degli algoritmi di ottimizzazione strutturale e geometrica. Con la prima si intende il processo al cui termine si riesce ad avere la maggiore performance strutturale a parità di peso: nel caso in questione sono stati utilizzati algoritmi genetici e processi iterativi. L’ottimizzazione geometrica è consistita nella ricerca della fabbricabilità dell’opera, nella riduzione della curvatura media dei pannelli di vetro e nella successiva pannellizzazione della superficie. Infine, grazie al codice creato, è stato possibile esportare facilmente tutti i dati ed eseguire tutte le analisi sfruttando i software più indicati per ogni esigenza.
Abstract
This thesis is built around the conception, development and optimization of pedestrian bridge, also known as Skywalk. The project
The structure was modelled entirely on Grasshopper, a Rhino3D plug-in which allows
to parametrically design objects. This programming environment allowed to interface add-ons, both proprietary and made on purpose by the author. With those tools, the author was able to develop algorithms for structural and geometric optimization. The former consisted in the process at the end of which the best structural performance is found, at constant weight: in the case studym genetic algorithms and iterative processes were used. Geometric optimization consisted in seeking the fabricability of the architecture, in the reduction of overall panel curvatures and eventually in the panelization of the surface. Finally, thanks to the code so created, it was possible to easily export all data and run required analysis exploiting the software more suited for each needing.
Introduction
Scope of the work Modern architecture brought innovative ways to approach the field. New materials, integrated workflows, complex structures, energetics, structural optimization and new shapes, specifically free forms. In such landscape in constant evolution, computational and design tools must keep up the pace, as long with designer’s in-depth knowledge. In this thesis, a toroidal Skywalk is designed from scratches. In order to do so, several problems arised.
The first problem of course, is to draw such a complex surface and the relative structure. Commonly used tools, such as AutoCAD or Revit, are not suited enough. Furthermore, it is well known that projects change hundreds of times during their conception and even their realization: updating it with CAD and subsequentely with all the softwares involved (for structural analysis, energetics, fabrication, costs, etc...) it is unfeasible.
Those architectures are highly expensive, thus every effort should be conducted to reduce costs. This practice is commonly refered to as optimization, since the aim is optimizing the use of material or the fabrication (both related to costs) to have best results with least expenses. Certain assemblies are expensive beyond imagination or not even possible to realize. In this case, the geometry itself must be modeled to make feasible the unfeasible.
For what concerns fabricability, the project presents a lot of different nodes, this again should catch the engineer’s attention and make him struggle to find a way to fabricate them in series to drive costs down.
For what concerns the project as a whole, the interoperability between softwares is crucial. As aforementioned, in the workflow of free form architectures a lot of softwares find the way in. Models must pass seamlessly from one software to one other, without loss of information and within short time. Extremizing the concept, the best would be found if the entire project, with all the information it carries, could flow from one step to the next without any external input required. The only way do so, is to have the whole thing written in a parametric fashion, so that when an input is changed, the whole projects with its various outputs updates.
Contents In this work, all the major issues above stated are faced. In particular, the work presents a full parametric geometry definition, both structural and geometrical optimization, where the former is dealt with in two different approaches.
Geometry was defined mostly using Grasshopper and Rhino3D for visualization. Early study on the spiral were conducted in MathCad.
Structural optimization was done using Genetic algorithms, Evolutionary Multi-Objective (EMO) algortihms and through permutation. The three procedures and some background are given in the present work.
Geometry optimization primarely consisted in reduction of average panel curvature and panelization. Panelization is the term commonly used to refer to the procedure through
which a continuous surface is reduced in a number of discrete panels, which are to be produced by an even smaller number of molds to keep costs down.
The structure was first analyzed for both static and seismic load. Some loads, such as wind pressure and snow load, are not immediate to define given a such uncommon shape, thus requiring a special script to calculate them.
Several softwares were used, thus needing data to flow from one to the other, and this was the most arduous part. From Grasshopper to GSA, Excel, SAP2000, and from MathCad, MatLab, C# to Grasshopper.
Contents
I Architectural Conception
1 The state of art……….………..……...15
1.1 The Grand Canyon Skywalk……….………...15
1.2 The Glacier Skywalk………17
2 Coding in Grasshopper………19
2.1 Component………..19
2.2 Custom Component………22
II Topology Definition
3 The Torus…………..………273.1 Mathematics of Elliptic Torus……….27
3.2 Principal Curvatures………28
3.3 Mean Curvature………...29
3.4 Gaussian Curvature……….30
3.5 Geodesic Curve………31
III Structural Behavior
4 Arch on space…………...……….37 5 Toroidal Shell………..………..39 5.1 Stress Behaviour……….…..39 5.2 Strain behaviour……….…..40 5.3 Principal Stress……….…42IV Spiral Layout
6 Chapter Genesis of the spiral………47V Structural Optimization
7 Optimization using Genetic Algorithms………..………...53
7.1 Pros and Cons………..54
7.2 The Process………...55 7.3 Fitness Functions……….57 7.4 Selection Mechanisms……….61 7.5 Coupling Algorithms………...62 7.6 Coalescence Algorithms………..64 7.7 Mutation Factories………...65
7.8 Genetic Algorithms applied to structural optimization……….66
7.9 Beam optimization………...68
7.9.1 Goal: beam utilization factor with non-linear weighted-average………..70
8 Evolutionary Multi-Criterion Optimization………..…….71
8.1 Introduction……….71
8.2 Optimal Control………...74
8.3 Pareto Front……….75
8.4 Butter and Guns example………76
8.5 Weak Pareto Efficiency………77
8.6 EMO applied to structural optimization……….79
8.7 Spiral layout optimization………...81
8.8 Permutation……….84
VI Panelization
9 Glass panel and panelization………..………..969.1 introduction……….96
9.2 Creating the mold………96
9.3 Zebra analysis………..………...99
VII Structural Design
10
Chapter………...……….Err
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10.1
Material………...Err
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10.1.1
Steel………Err
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10.1.2
Glass………...Err
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10.2 Load
Analysis……….Errore. Il
segnalibro non è definito.
10.2.1 Static
Loads………Errore. Il
segnalibro non è definito.
10.3 Steel
members………Errore. Il
segnalibro non è definito.
11 Chapter………105
11.1 Steel members………105
11.2 Glass panel………..105
11.3 Buckling analysis………108
PART I
1 Chapter
The state of art
1.1 The Grand Canyon Skywalk
Figure 1-a Grand canyon skywalk
The Grand Canyon is the Skywalk is located on a magnificent cliff overlooking the western edge. Opened in 2007 on the Hualapai Indian Reservation about 120 miles (193 km) from Las Vegas, the unique U-shaped “glass bridge” is a partially enclosed walkway that cantilevers 21 meters out above a 219 meters high cliff. This “bridge to nowhere” is a spectacular bit of engineering even if it does not fit everyone’s definition of what a bridge is. If categorized as a pedestrian footbridge, the Skywalk is the highest of its type in the world.
Figure 1-b top and perspective view of the skywalk
The walkway itself is composed of two parallel box beams each 2 meters deep and 81 cm wide. The width of the walkway is 3 meters as measured between the two glass railings that are each 2 meters high. The glass floor is 7 cm thick and is made of five layers of glass. To prevent
scratches on the top layer, visitors are required to wear shoe slippers. The structure can withstand an 8.0 earthquake and is supported on tuned mass dampers to reduce normal vibrations from the wind and people. The U-shaped structure is supported on 8 box column posts 32 inches square. The footings are supported on foundations 14 meters deep. Built at a cost of 8 million dollars, the entire project is the brainchild of Las Vegas businessman David Jin. The design team consisted of Las Vegas architect Mark Johnson of M.R.J. Architects and Lochsa Engineering. APCO Construction was the contractor and RWDI of Ontario did vibration and wind test studies.
The Skywalk is located on a side canyon that leads down to the Colorado River about 2.4 km away. There is an elevation difference of 1100 meters between the Skywalk and the river though the vertical height of the Skywalk is only 219 meters above the plateau directly below it.
1.2 The Glacier Skywalk
Figure 1-c Glacier Skywalk
The glass walk is a self‐anchored, eccentric suspension structure. The solid part of the cantilever is composed of two trapezoidal steel box girders created from custom-formed metal plates that are welded together and anchored to the mountain with a series of high-strength steel tendons that vary in length from 10 to 20 m.
A draped suspension cable runs along the outside face of the inner guard rail. To counter the eccentricity of this support, the walk is curved in plan to generate a horizontal couple that cancels out the vertical couple. This system minimizes the amount of structure that is visible to visitors, particularly beneath the glass floor of the skywalk.
Steel tubes that were custom-bent in three dimensions support the glass portion of the walkway. The tubes contain cables similar to those used in large cable-stayed bridges. Snow, wind, and pedestrian vibration loading were the prime considerations, largely due to the inherent flexibility of the structural system selected. The snow load used for design is twice what was used for pedestrian loading, and vibration due to both wind and pedestrian action were both identified as potential challenges for this structure.
Wind vibration is mitigated by wind deflectors located atop the outer glass guardrail, and four tuned-mass dampers are located beneath the outer glass rail, below floor level, to temper pedestrian-induced vibrations. However, the design team wanted visitors to experience the skywalk at a visceral level so the dampers temper, but do not eliminate the wobble effect that you get when large numbers of people are walking on the skywalk. Designing for dynamic and unbalanced loading was the biggest structural design challenge. The issue with unbalanced
loading, when you have people on only a portion of the glass walk, is that the system is designed and optimized for uniform loading. Under uniform loading, all of the main structural members act primarily as axial members, with little bending or torsional forces. Under unbalanced loading, bending and torsional forces become much more dominant and this results in increased flexibility and deflection.
To resolve these issues, the design team had to carefully design details for all of the possible unbalanced loading scenarios. In particular, they had to be certain that the glass floor panels could undergo the deflection associated with unbalanced live loads. The end design utilizes the high torsional stiffness of the primary, hollow structural-section compression member along with contributions from a number of secondary framing elements.
The glass system used in the walkway includes four layers. The bottom three layers are structural glass that has been tempered and heat-strengthened. These are topped with a fourth layer of sacrificial glass panels that create the surface on which visitors walk. The top sheet is a surface treatment, and if it does get scratched over time, or if a piece breaks, then the top surface is easily changeable. The design team wanted a system that would enable visitors to wear their own footwear—something that is not possible at the Grand Canyon's observation walkway. The materials for the skywalk were chosen to blend into the cliff side. While there was never any pretention that it should be anything but something that was man-made, it should be—within that context—as subtle and as sensitive a fit as possible to the mountainside. To meet that goal, a minimal number of materials, and materials that would wear well without the need for protective coatings, were chosen. These included weathering steel that can be left exposed to the elements without deteriorating, glass, and wood in locations at which visitors will pause to enjoy the view or rest. Although the Glacier Skywalk was made almost entirely of prefabricated pieces, construction took more than two years.
2 Chapter
Coding in Grasshopper
Complex surfaces, freeform geometry and relative structures are difficult to draw and commonly used tools, such as AutoCAD or Revit, are not suited enough. Furthermore, it is well known that projects change hundreds of times during their conception and their realization: updating it with CAD and subsequenteky with all the softwares involved (for structural analysis, energetics, fabrication, costs, etc…) it is unfeasible.
One of the software that allow to satisfy this demand is Grasshopper is a plug-in for the highly advanced 3D modeler Rhinoceros. Grasshopper allows the user to conceive all its design by parametric means. This allow the user to see and exploit all the possible solutions inside the domain space. In other words, the inputs (the parameters) vary in a defined domain. The co-domain is the actual design, linked to the set multidimensional co-domain through the user-made script (all those little blocks interlinked one to each other). Thus, is very important to having rightly set the domain and the function, because the output depends exclusively on those ones. Grasshopper hosts a lot of different plug-ins, each one specialized on some aspect of the design workflow. Numerous plug-ins have been used throughout the entire project development. In the following, a list of them is presented along with a short introduction.
2.1 Component
Karamba Karamba is a plug-in by Bollinger+Grohmann ingenieur, a German structural engineering firm. It allows to perform structural analysis, right inside grasshopper.
Although it’s not as powerful as other stand-alone well-known softwares, first of all SAP2000 and GSA, its being directly connected to the design workflow increase the time performance. Moreover, since it’s inside grasshopper, it can perform various operations at any step of the workflow, something which would be quite difficult to achieve with a standalone software. Here’s a list of some interesting operations Karamba is able to do:
• Actively operate on the design geometry (topology);
• Modify any set of data, according to certain structural output;
• Allowing the creation of an optimization loop, with the aim of minimize (maximize) of some structural output.
Geometry Gym Geometry Gym is another fundamental plug-in for anyone interested in structural analysis/optimization, keeping linked with Grasshopper. It is composed by two major parts.
The first one, named Bullant, simply gives some new powerful design tools, which span from simple geometric boolean operations, such as extrapolation of closed cells from a curve network, to more sophisticated ones, such as minimum surfaces, geodesic domes and tessellation.
The second one, basically enables reading/writing of files from/to any structural software (e.g. SAP2000, GSA, Robot Structural Analysis). This way, projects realized inside Grasshopper can be exported, analyzed and re-imported into it, giving the designer an important edge over the workflow process.
Galapagos Galapagos is a plug-in already implemented inside Grasshopper. It gives the possibility to use genetic algorithms, directly inside grasshopper.
Basically, it allows to minimize (maximize) a target function F(xi) of an unlimited number of variables (genes). All the n genes have a domain, so the result is an n-dimension space, with the function F(xi) leading to a 1-dimension co-domain, which target value (an integer, of course) is the being optimized. Each set of n different genes, which is a n-dimension vector, is called chromosome. Best fitting chromosomes, the ones whose co-domain’s value is the lowest, are made mate to each other, creating a supposedly better offspring. After a while, the best member in the final population can be chosen , which eventually will have the lowest co-domain value, and the process can be halted. In this thesis, initially it was used to perform the global structural optimization, but was later abandoned in favour of a multi-objective genetic algorithm.
LunchBox This plug-in has pretty much everything inside, from unusual geometry design, to panels and structures. The tools I find mostly useful are situated under the workflow bar. There one can find tools improving the design workflow, such as write/read a .xls file, bake, launch an application, run a rhino command.
Octopus Octopus is seen as an extension to Galapagos by introducing multiple fitness values to
the optimization. The best trade-offs between those objectives are searched, producing a set of possible optimum solutions that ideally reach from one extreme trade-off to the other uniformly. Results are shown in a two- to five-dimensional, 3D-navigable solution-viewport. User-preferences on specific solutions can be specified when the optimization is paused, so later genomes will be searched near the selection. The history of the evolutionary runs can be recorded, saved, and loaded again.
• SPEA-2 multi-objective evolutionary algorithm by Zitler, Deb, Thiele at ETH Zurich • Arbitrary number of objective to optimize
• Integration of User-Preferences for a biased search
• Recorded history 3D solution space with abstraction to 5D-objective problems, navigable
• Save the current state directly within the Grasshopper document
For now it is based on the Improved Strength Pareto Evolutionary Algorithm SPEA-2 algorithm by Zitzler, Deb, Thiele at ETH Zürich and David Rutten’s Galapagos User Interface. It is developed by Robert Vierlinger in cooperation with Christoph Zimmel, karamba3d.com and Bollinger Grohmann Schneider ZT GmbH Vienna.
Hoopsnake Hoopsnake came to solve one of the biggest flaw of Grasshopper: the impossibility to perform a so called for loop. Since Grasshopper has a linear workflow, meaning that the flow of data goes in one and one only direction (hence the plug position of any the cluster, and the link shape between them). Hoopsnake comes to change this habit. There one can input the initial data, at step 0, run them through the script, and re-input the updated data from a different input plug. As any loop, there must be some kind of counter which stops the process whenever it reaches a certain threshold.
WeaverBird WeaverBird is a powerful mesh editing tool. It can perform various mesh subdivisions (e.g. Catmull-Clark, Sierpinsky, midedge, Loop). It is really helpful when we want perform accurate finite element analyses.
Kangaroo Kangaroo is probably the most known and used plug-in of all. What it does is to add a physical engine and number of physical forces and interactions. A physical engine may have a lot of useful uses. In this thesis it has not been used.
Evolute Evolute is a plug-in for Rhino, not for Grasshopper, differently from the others. This
software is an advanced geometry optimization tool for freeform surfaces with a user friendly interface. Established computational tools from Evolute's core software library as well as ground breaking technology from our cutting edge research results provide you with optimization functionality not offered by any other CAD system.
This software offers:
• multi-resolution mesh modelling; • global and local subdivision rules;
• mesh editing tools and mesh optimization for various goals (closeness, planarity, fairness, coplanarity, edge length repetition, ballpacking);
• specification of vertices as anchor/corner points, constraints (floor slabs, generai co-planarity constraints, reference curves, fine grained fairing);
• specific analysis such as closeness, planarity, edge length, principal curvature and asymptotic line analysis;
• pattern mapping; • NURBS fitting.
2.2 Custom Component
Several custom clusters have been created to solve certain problems. By my point of view, what distinguishes a cluster by a common grasshopper script file, is the huge amount of components/scripting parts, compared with the number of inputs and outputs.
Pre-stress Force
This Cluster is conceived to calculate the resultant of a prestress load on a curve element. The cluster calculate the curvature at each point of a curve and afterwards we need just to make a multiplication with the prestress force.
Figure 2-a Pre-stress Load
Moment of inertia
Grasshopper has a component that calculate the inertia moment of a generic section. Have a double check is always better and this cluster return the correct value.
PART II
3 Chapter
The Torus
3.1 Mathematics of Elliptic Torus
A Torus is a surface of revolution generated by revolving a circle about an axis coplanar with the circle. Let the radius from the center of the hole to the center of the torus tube R, and the radius of the tube be r. Then the equation in Cartesian coordinates for a torus azimuthally symmetric about the z-axis is:
An Elliptic Torus instead is a generalization of the Torus which is produced by rotating an ellipse having horizontal semi-axis r1,vertical semi-axis r2 and located a distance R away from
the rotational axis. The parametric equations are:
for u, v ∈ [0, 2π].
Figure 3-a Torus shape
Three types of torus, known as the standard tori, are possible, depending on the relative sizes of r1 and R. R > r1 corresponds to the ring torus, R = r1 corresponds to a horn torus which is
Figure 3-b ring torus (left) horn torus (center) spindle torus (rght)
3.2 Principal Curvatures
Before we can introduce principal curvature directions we have to introduce the concept of normal curvature. Let S be a surface in R3 that is given by the graph of a smooth function f(x,
y). Assume that S passes through the origin p and its tangent plane in p is represented by the {z = 0} plane.
Let N = {0, 0, 1} be a unit normal to S at the origin.
Figure 3-c Normal curvatures of a surface S at a point p
Furthermore we denote v = {v1, v2, 0} a unit vector in TpS. Let c be the parameterized
curve given by slicing S through the plane spanned by v and N. We obtain, , , ,
For a plane curve c we introduce the concept of signed curvature k at p with respect to the unit normal N: kv is the reciprocal of the radius of the osculating circle to c at p, taken
with sign as in the examples below:
The maximum and minimum of the normal curvature k1 and k2 at a given point on a surface
are called the principal curvatures.
Figure 3-e Line of principal curvature through two point
The principal curvatures measure the maximum and minimum bending of a regular surface at each point. The Gaussian curvature K and mean curvature H are related to k1 and k2 by
This can be written as a quadratic equation
Which has solutions
3.3 Mean Curvature
Let k1 and k2 be the principal curvatures at a point on a surface, then their mean is called the
Let R1 and R2 be the radii corresponding to the principal curvatures, then the multiplicative
inverse of the mean curvature in given by
In terms of the Gaussian curvature K,
3.4 Gaussian Curvature
Let k1 and k2 be the principal curvatures at a point on a surface, then their product is called
The Gaussian curvature K and mean curvature H satisfy
with equality only at umbilic points, since
An umbilic point is a point on a surface at which the curvature is the same in any direction and every tangent vector is a principal direction.
3.5 Geodesic Curve
A geodesic is a locally length-minimizing curve. Equivalently, it is a path that a particle which is not accelerating would follow. In the plane, the geodesics are straight lines. On the sphere, the geodesics are great circles.
Geodesics preserve a direction on a surface and the normal vector to any point of a geodesic arc lies along the normal to a surface at that point.
Figure 3-f geodesic curve (red) on a surface (grey).Blue vector represent the normal vector of curve, Green vector is the normal of a surface
Recall our parameterization of the torus:
From O’Neill, a geodesic can be parameterized
√ √ √
If we could integrate this, we’d have a formula for u in terms of v:
cos cos
This integral has no closed form solution but the formula for du/dv depends on only one parameter, h, the geodesic’s slant which measures of the angle between the geodesic and the curves.
The possible values of h gives us an idea of the different kinds of geodesics that exist on the torus. The term under the radical must be real. This allows us to classify the possible geodesics into several families.
h=0. These are the meridians:
Figure 3-g Meridian geodesic
An intuitive way to see that meridians are geodesics is to realize that the torus has a mirror symmetry through meridians. Anything that would push the geodesic off a meridian in one direction is balanced on the opposite side, so a geodesic that starts on a meridian cannot leave
it. A similar argument can be made for both the inner and outer equator, which means they must be geodesics as well.
0 < |h| < c-a. These geodesics cross both the inner and outer equators. We call these geodesics
unbounded, because they can pass through all points on the surface. These geodesics cross the inner and outer equators at different angles.
Figure 3-h Unbounded geodesic
h = c-a. As h approaches this value from below, the angle a geodesic makes with the inner
equator approaches zero. Hence when h = c-a, one geodesic is the inner equator.
Figure 3-i Inner equator geodesic
c – a < |h| < c + a. We call these geodesics bounded. Here is one of the simplest bounded
geodesics, which touches each barrier curve once.
h = c + a. As h approaches c + a, the barrier curves approach the outer equator. Hence the one
geodesic for which h = c + a is the outer equator.
PART III
4 Chapter
Arch on space
The semicircular beam carry an uniform load q and it has translation and rotation fixed at support.
The moment in a generic section are
Then, from the equilibrium differential equation we get
traslation and rotation in the middle section section are
Figure 4-a Shear stress (left); Bending moment (center); Torcent moment (right)
The trend of cross section forces along the beam are similar of the double encastred beam at support but because of the spatial curve in this case we see a torcent moment due from the eccentricity of the acting load.
The structural behaviour of the arch beam on space has a behaviour very close to that of the beam encastred at the support. Because of the spatiality, we find a torque stress along the route. The supporting area is characterized by a tense area at upper edge and compression area on inferior area. At centerline the behaviour is reverse.
Figure 4-c various shape of different skywalk
The parametric structural model allow us to show in which way the stress of the beam change making change in the initial boundary tangent condition of the skywalk..
The shear stress obviously keep the same value while the bending moment and torcent moment get higher.
5 Chapter
Toroidal Shell
5.1 Stress Behaviour
For better understand the flow of the force we have moved to analyse the continuous body in order to have a more accurate idea of the possible stress and strain behaviour of the shell. The structural element has proved to be very sensitive to discretization into shell elements and to overcome this problem we adopted a 25 cm edge triangular mesh.
The structural behaviour has analogy with the beam element, but it has an unexpected behaviour in the middle. The center line presents technical prescriptions of the compressive stress and tensile both the upper and the lower area. The prevailing force are those in analogy with the beam element but is definitely not to overlook this alternation of sign in the signs.
Figure 5-a principal stress tension
5.2 Strain behaviour
The strain behavior of a structure is closely related to stress behavior through the constitutive law. For each point on the surface it has been identified a displacement vector that represents precisely how much and in what direction an infinitesimal element has moved. The elements with the same amplitude have been connected to each other generating the contour lines. This type of information is helpful to better understand the behavior of the structure but it has not been used to generate the resistant structure.
Figure 5-c iso displacement line
5.3 Principal Stress
The study of continuous body allows us to discover at every point the stress tensor.
The stress tensor is a field tensor – it depends on factors external to the material. In order for a stress not to move the material, the stress tensor must be symmetric: σij=σji.
The general form is thus:
Or, in alternative notation,
The general stress tensor has six independent components and could require us to do a lot of calculations. To make things easier it can be rotated into the principal stress tensor by a suitable change of axes.
The magnitudes of the components of the stress tensor depend on how we have defined the orthogonal x1, x2 and x3 axes. For every stress state, we can rotate the axes, so that the only non-zero components of the stress tensor are the ones along the diagonal:
that is, there are no shear stress components, only normal stress components.
The principal stresses are the eigenvalues of the stress tensor. These can be found from the determinant equation:
This determinant is expanded out to produce a cubic equation from which the three possible values of λ can be found; these values are the principal stresses and they are the eigenvalue of the matrix. Instead, the eigenvector are the principal stress direction, so that direction with no shear stress.
PART IV
Spiral layout
6 Chapter
Genesis of the spiral
The parametric design allows us to see in real time the changes varying the parameters. Be able to descrive all the possible enveloping curves the toroidal shape allows us to explore all the possible solutions and therefore eligible to have all the possible outcomes of the particular problem. It was implemented a logic to describe in a few parameters the winding spiral. Imagine to dissect the toroidal shape with vertical plane. The derived geometry is the one described by the red curve and we denote by v. Now, we imagine to dissect the toroidal surface with a horizontal plane. The derived geometry is the one described by the blue curve and we denote by u. The pair of u and v numbers identify the point in common of these two curves and varying the parameters, one can identify all the horizontal and vertical sections of the surface.
Figure 6-a Vertical section curve (left); Horizontal section curve (blue)
Then a meridian is characterized by point with constant u and variable v. This argument allows us to draw any curve in uv space and consequently draw curve in xyz space. The following figures show the various steps followed to arrive at such result.
PART V
7 Chapter
Optimization using
Genetic Algorithms
The term Genetic Algorithm or Evolutionary Algorithm (EA) stands for a class of stochastic optimization methods that simulate the process of natural evolution. The origins of EAs can be traced back to the late 1950s, and since the 1970s several evolutionary methodologies have been proposed, mainly genetic algorithms, evolutionary programming, and evolution strategies [?]. All of these approaches operate on a set of candidate solutions. Using strong simplifications, this set is subsequently modified by the two basic principles: selection and variation. While selection mimics the competition for reproduction and resources among living beings, the other principle, variation, imitates the natural capability of creating ”new” living beings by means of recombination and mutation. Although the underlying mechanisms are simple, these algorithms have proven themselves as a general, robust and powerful search mechanism. In particular, they possess several characteristics that are desirable for problems involving:
• multiple conflicting objectives,
• intractably large and highly complex search spaces.
As a result, numerous algorithmic variants have been proposed and applied to various problem domains since the mid-1980s. The rapidly growing interest in the area of multi-objective evolutionary algorithms (MOEAs) is reflected by, e.g., a conference series and some books. The following section, 9, focuses on algorithm design issues and presents concepts and techniques that have been developed to deal with the additional complexity caused by multiple objectives. These issues will be illustrated on the basis of a specific algorithmic variant, namely SPEA2.
How does a Genetic Algorithm works? The baseline is simple. Every input has a domain range, the set of n inputs creates a n dimension domain. The algorithm creates a starting population evenly distributed along the whole n-dimension domain, then each member is linked to a value given by the goal equation. Highest value members are preserved and combine to each other giving offspring, while the others are discarded.
Genetic algorithm were recently implemented in Grasshopper, as a built-in tool, named Galapagos, which is an easy-to-use genetic solver.
7.1 Pros and Cons
PROS For how little are used, all is not bleak and dismal however. Evolutionary Algorithms
have strong benefits, some of them are even rather unique amongst the plethora of computational methods. They are remarkably flexible for example, able to tackle a wide variety of problems.
Evolutionary Algorithms will happily chew on problems that have been under- or over-constrained or otherwise poorly formulated. Even a pre-maturely aborted run will yield something which could be called a result. It might not be a very good result, but it will be a result of sorts.
Genetic Algorithms don’t usually need too much tweaking to work. Of course setting up parameters correctly, would strongly improve the time needed for the algorithm to converge, but it’s not mandatory. Traditional algorithms need to have integration step and, generally speaking, parameters set correctly otherwise they are not going to converge, never. They may be even converge to other solutions, which are not the good one, thus being dangerous to use by unexpert users.
Finally, run-time process is highly transparent and browsable, and there exist a lot of opportunities for dialogue between algorithm and human. The solver can be coached across barriers with the aid of human intelligence, or it can be goaded into exploring sub-optimal branches and superficially dead-ends.
CONS Firstly, Evolutionary Algorithms are very slow since they follow multiple paths at the
same time, and they may happen not to be the best ones at all. Each optimization process may run even for days. Since each iteration need to be run for each different genome, it is compulsory make them as fast as possible. For instance, a light/shadow or acoustic computation for example may easily take a minute per iteration. Supposing there are needed at least 50 generations of 50 individuals each (just to keep them low), the full computation will take about two days. Secondly, evolutionary Algorithms do not guarantee a solution. Unless a predefined good-enough value is specified, the process will tend to run on indefinitely, but this may be a common problem for any kind of algorithm. Computation time can vary a lot depending on the variables, solution space, parameters, goal function and so on. This means that an expert user should bring in mind to manage a lot of different issues for optimizing the algorithm. Other algorithms may have less configuration freedom and less running time required.
7.2 The Process
Figure 7-a Genetic Algorithm population
What you see here is the Fitness Landscape of a particular model. The model contains two variables, meaning two values which are allowed to change. In Evolutionary Computing we refer to variables as genes. Let’s say Gene A change: the fitness of the entire model goes up or down. But for every value of A, it is also possible to vary Gene B, resulting in better or worse combinations of A and B. Every combination of A and B results in a particular fitness, and this fitness is expressed as the height of the Fitness Landscape.
A model with n genes would be a n-dimensional fitness volume deformed in n + 1 dimensions instead of a two-dimensional fitness plane deformed in 3 dimensions. The initial step of the solver is to populate the landscape (or "model-space") with a random collection of individuals (or "genomes"). A genome is nothing more than a specific value for each and every gene. In the above case, a genome could for example be A = 2 B = 13. The solver will then evaluate the fitness for each and every one of these random genomes, giving the distribution shown in Figure 7-a-i. Once the fitness of every genome is known (i.e., the elevation of the red dots), it is possible to make a hierarchy from fittest to lamest. Lowest fit genomes are eliminated while the others are preserved, as shown inFigure 7-a-ii. Now, the fittest genome is very unlike to be the
searched solution, since all of them were randomly selected. On the contrary, more evolution is needed with further generations. To do so, the best performing genomes in Generation 0 breed to create Generation 1. Two genomes breed, their offspring will end up somewhere in the intermediate model-space, thus exploring fresh ground as shown in Figure 7-a-iii. Repeating this process results in having populations less and less chaotic, which moving forward to cluster around the three peaks, representing the function maximum in the given genes domain. The algorithm halts when a certain generation is reached, leaving the user the job to evaluate the result. In order to perform this process, an Evolutionary Solver requires five interlocking parts, which will be discussed in the following. Let us enumerate them:
• Fitness Function • Selection Mechanism • Coupling Algorithm • Coalescence Algorithm • Mutation Factory
7.3 Fitness Functions
A fit individual is on average able to produce more offspring than an unfit one, so we could say that fitness equals the number of genetic children. A better measure yet would be to count the number of grand-children. And a better measure yet would be to count the allele frequency in the gene-pool of the genes that made up the individual in question. At least in Evolutionary Computation, fitness is a very easy concept. Fitness is whatever it is set to be by the user. If for example we are seeking to position a shape so that it may be milled with minimum material waste, there is a very strict fitness function that leaves no room for argument.
Let’s have a look at the fitness landscape again and let’s imagine it represents a model that seeks to encase an object in a minimum volume bounding-box. A minimum bounding-box is the smallest orthogonal box that completely contains any given shape. In the image below, the green shape is encased by two bounding boxes. B has a smaller area than A and is therefore fitter, see Figure 7-b-iii.
Figure 7-b Path to optimum point
Let’s say we manage to pick a random genome that is at the bad end of the fitness scale, i.e. at the bottom of the fitness landscape. What is possible to say about the blood-line of this genome? When the descendants of a particular genome are tracked, there is always a large
amount of randomness involved due to the workings of the Solver, but there is a strong general tendency that can be distinguished. Just like water will always flow downhill along the steepest slope, so genetic descendants will generally climb uphill along the steepest slope, as in Figure 7-b-i. Every individual tries to maximize its own fitness, as high fitness is rewarded by the solver. And the steepest uphill climb is the fastest way towards high fitness. So if the black sphere represents the location of the ancestral genome, the orange track represents the pathway of its most successful offspring. Repeating this exercise for a large amount of sample points will show something about how the Solver and the Fitness Landscape interact, Figure 7-c. Since every genome is pulled uphill, every peak in the fitness landscape has a basin of attraction around it. This basin represents all the points in model-space that will converge upon that specific peak. It is important to notice that the area of the basin is in no way representative of the quality of the peak. Indeed, a very poor solution may have a large basin of attraction while a good peak might have a small catchment area. Problems like this are typically very difficult to solve, as the solution tends to get stuck in local optima. Problematic fitness functions will be discussed later on (see, Figure 7-e).
First, let’s have a closer look at the actual fitness landscape (Figure 8.2.3a) for minimum bounding-box model. It is remarkable how complicated results a fitting function for such a problem which at first sight seems relatively easy. The x-axis rotation is mapped along the Gene A direction and the y-axis rotation along the Gene B direction. Hence, every point on the AB plane represents a unique rotation composed of two angles. The elevation of this point is a direct mapping of the volume of the bounding-box at those two rotation angles.
Figure 7-c Landscape views
The first thing to notice is that the landscape is periodic. I.e., it repeats itself every 90 degrees in both directions. Also, this landscape is in fact inverted as we’re looking for a minimum volume, not a maximum one. Thus, the orange peaks in fact represent the worst solutions to this problem. Note that there are 16 of these peaks in the entire range and that they are rounded. A view from the bottom up is shown in Figure 7-c.
It would appear that the lowest points in this landscape (the minimum bounding boxes) are both fewer in number and of a different kind: there are only 8 optimal solutions and they are all very sharp, indicating a somewhat more fragile state. All the solutions are of equal quality and there are no local optima at all. It is possible to generalize this landscape to a 2-dimensional graph, as in Figure 7-e.
Figure 7-d Co-domain
No matter where you end up as an ancestral genome, your blood-line will always find its way to a minimum bounding box. There’s nowhere for it to get stuck. So it’s really just a question about who gets there first. If we look at a slightly more complex fitness graph, it becomes apparent that this need not be the case: This fitness landscape has two kinds of solutions. The high quality sharp ones near the bottom of the graph and the low quality flat ones near the top. The basin of attraction is given for both solutions (yellow for high quality, pink for low quality) and you can see that about half of the model space is attracted to the low quality solutions. An even worse example (flipped upright again this time, so high values indicate good solutions) would be the fitness landscape given in Figure 7-e-ii. The basins for these peaks are very small indeed and therefore easy to miss by a random sampling of the landscape. As soon as a lucky genome finds the peak on the left, its offspring will rapidly populate the low peak causing the rest of the population to go extinct. It is now even less likely that the better peak on the right will be found. The smaller the basins for solution, the harder it is to solve a problem with an evolutionary algorithm.
Figure 7-e Difficult fitness landscapes
Another example of a cumbersome problem to solve would be a discontinuous fitness landscape as the one shown in Figure 7-e-ii.
Even though there are strictly speaking no local optima, there is also no improvement on the plateaus. A genome which finds itself in the middle of one of these horizontal patches doesn’t know where to go. If it takes a step to the left, nothing changes. If it takes a step to the right, nothing changes. There’s no pressure in this fitness landscape, so all the genomes will wander about aimlessly, until one of them has the good fortune to suddenly step onto a higher plateau. At this point it will quickly dominate the gene-pool and the wandering starts again until the next plateau is accidentally found. Even worse than this, though, is a landscape that has a high degree of noise or chaos. A landscape may be continuous and yet feature so much detail that it becomes impossible to make any intelligible pronunciations regarding the fitness of a local patch, as in Figure 7-e-iii.
In a landscape like this, the parents may both be very similar and both be very fit, but when they mate the offspring might end up in one of the fissures. A landscape like this defies navigation.
7.4 Selection Mechanisms
Figure 7-f Selection methods
Biological Evolution proceeds by Natural Selection, the ruthless force identified by Darwin as the arbiter of progress. Natural Selection affects the direction of the genepool over time by regulating who gets to mate. Using an Evolutionary Solvers require to set who gets to mate. In the following, the mechanism for parent selection used in Galapagos, are explained.
Isotropic Selection In Figure 7-f-i, you can see the isotropic selection. It’s the simplest kind of algorithm you can imagine, actually it is the absence of a selection algorithm.
In Isotropic Selection everyone gets to mate. No matter where you find yourself on this fitness graph, your chances of ending up in a mating couple are constant. You might think that this is a particularly pointless selection strategy as it does nothing to further the evolution of the gene-pool. But it is not without precedent in nature. Take for example wind-pollination or coral spawning. If you’re a sexually functioning member of such a species, you get to play ball come mating season. Another example would be females in a walrus colony. Every female in a colony gets to breed with the dominant male, no matter how fit or unfit she is. Isotropic Selection is certainly not without function either. For one, it dampens the speed with which a population runs uphill. It therefore acts as a safe-guard against a premature colonization of a local, or possibly inferior, optimum.
Exclusive Selection In Figure 7-f-ii, you can see the exclusive selection where only the top N% of the population get to mate. If one is lucky enough to be in the top N%, it’ll likely have multiple offspring. In fact, since only a small part of the population breeds, in order to create a new population of the same size as the former, the top N% will need to have a lot of offspring.
Biased Selection Finally, in Figure 7-f-iii, is sketched a biased selection, where the chance of
mating increases as the fitness increases. This is something we typically see with species that form stable couples. Everyone is basically capable of finding a mate, but the really attractive individuals manage to have more aspiring partners, thus increasing their chances of becomes genetic founders for future generations. Biased Selection can be amplified by using power functions, which have the effect of flattening or exaggerating the curve.
7.5 Coupling Algorithms
Coupling is the process of finding mates. Once a genome has been elected to mate by the active Selection Algorithm, it has to pick a mate from the population to complete the act. Galapagos performs a selection by genomic distance. Figure 7-g shows a Genome Map.
Figure 7-g Genome Map
There are displayed all the genomes (individuals) in a certain population as dots on a grid. The distance between two genomes on the grid is roughly analogous with the distance between the genomes in gene-space. I say roughly because it is in fact impossible to draw a map with exact distances. A single genome is defined by a number of genes. We assume that all the genomes in a species have the same number of genes (this is not technically a limitation of Evolutionary Algorithms, even though it is currently a limitation of Galapagos). Therefore the distance between two genomes is an N-Dimensional value, where N equals the number of genes. It is not possible to accurately display an N-Dimensional point cloud on a 2-Dimensional screen so the Genome Map is only a coarse approximation. It also follows that the axes of this graph have no meaning whatsoever, the only information a Genome Map conveys is which genomes are more or less similar (close together) and which genomes are more or less different (far apart).
Figure 7-h Type of range selection
That red dot in Figure 7-h Type of range selection is the specimen member. It could be possible to limit the search of potential partners to its immediate neighbourhood. This means that it will mate with individuals who are very much like it thus its offspring will also be very similar.
When this is taken to extremes we call it incestuous mating behaviour and it can become detrimental pretty quickly. In the digital world of Evolutionary Solvers the biggest risk of incest is a rapid decline in population diversity. Low diversity decreases the chances of finding alternate solution basins and thus it risks getting stuck in local optima.
The other extreme is to exclude everyone near you. At some point the genomes at the other end of the scale become so different as to be incompatible. This is called zoophilic mating and it can be equally detrimental. This is especially true when a population is not a single group of genomes, but in fact contains multiple sub-species, each of which is climbing their own little fitness peak. You definitely do not want to mate with a member in a different sub-species, as the offspring would likely land somewhere in the middle. And since these two species are climbing different peaks, "in the middle" actually puts you in a fitness valley. It would seem that the best option is to balance in-breeding and out-breeding. To select individuals that are not too close and not too far. In Galapagos one can specify an inbreeding factor (between
-100% and +-100%, total out-breeding vs. total in-breeding respectively) that allows you to guide this relative offset: Note that mate selection at present completely ignores mate fitness. This is something that needs looking into for future releases, but even without any advanced selection algorithms the solver still works.
7.6 Coalescence Algorithms
Once a mate has been selected, offspring needs to be generated. The biological process of gene recombination is horrendously complicated and itself subject to evolution (meiotic drive for example). The digital variant is much more basic. This is partially because genes in evolutionary algorithms are not very similar to biological genes. Genes in evolutionary solvers like Galapagos behave like floating point numbers, that can assume all the values between two numerical extremes. When we mate two genomes, we need to decide what values to assign to the genes of the offspring. Again, Galapagos provides several mechanisms for achieving this. Imagine we have two genomes of four genes each. The combination of M and D is potentially a completely symmetrical process. A mechanism that is somewhat synonymous with biological recombination is Crossover Coalescence.
Figure 7-i Chromosome interaction
Crossover mating here each offspring inherits a random number of genes from parents. In this mechanism gene value is maintained.
Blend Coalescence will compute new values for genes based on both parents, basically averaging the values: It is also possible to add a blending preference based on relative fitness. If
one parent is fitter than the other for example, the former gene values will be more prominent in the offspring
7.7 Mutation Factories
All the mechanisms discussed so far (Selection, Coupling and Coalescence) are designed to improve the quality of solutions on a generation by generation basis. However all of them have a tendency to reduce the bio-diversity in a population. The only mechanism which can introduce diversity is mutation. Several types of mutation are available in the Galapagos core, though the nature of the implementation in Grasshopper at the moment restricts the possible mutation to only Point mutations. Before we get to mutations though, let us make a brief introduction on Genome Graphs. A popular way to display multi-dimensional points on a two-dimensional medium is to draw them as a series of lines that connect different values on a set of vertical bars. Each bar represents a single dimension. This way we can quite easily display not just points with any number of dimensions, but even points with a different number of dimensions in the same graph:
Figure 7-j Mutuation examples
Here for example we have a genome consisting of 5 genes. This genome is thus a point in the 5-dimensional space that delineates this particular species. When G0 is drawn at 1/3 , it means
that the value is one-third between the minimum and maximum allowed limits. The benefit of this graph is that it becomes quite easy to spot subspecies in a population, as well as lone individuals. When we apply mutations to a genome, we should see a change in the graph, as every unique genome has a unique graph.
Point Mutation Occurs when a single gene value is changed. This is currently the only
mutation type that is possible in Galapagos.
Inversion Mutation Where two adjacent gene value are swapped. They are only useful when
subsequent genes have a very specific relationship. It tends to drastically modify a genome and thus in most cases also drastically modify fitness. This is almost always a detrimental operation.
Addition and Deletion mutations Two examples of mutations that cannot be used on a
species which requires a fixed number of genes are .
7.8 Genetic Algorithms applied to structural optimization
Using Galapagos Genetic Algorithm, in connection with Karamba structural analyzer, gives the design workflow a strong edge over the so long looked for optimization loop (design-evaluation-optimization). The optimization works in this way:1. The genes are the beam cross section widths (height is fixed and is the same for everyone).
2. Thus, each chromosome describes all the section widths of the structure.
3. For each chromosome, Karamba’s engine solves the structure, finding the utilization ration of each beam.
4. For each generation, the top 10% is preserved, and a mutation factor of 5% is allowed. 5. Goal function may vary. In fact, since it’s not the final solution we’re looking for, one
can use different chromosome reward criteria: the goal value can be given as a) The sum of each beam’s utilization factor
b) The sum of a non-linear function of each beam’s utilization factor c) The overall failure percentage
7.9 Beam optimization
A first optimization has been conducted directly in Grasshopper. First of all, it should be defined what is meant by optimization, what are the inputs and the outputs. By optimization it’s meant the process with which, changing input values, the goal function is minimized (or maximized).
Input The input parameters are:
• The Number of starting spiral beams • The number of oscillation along the path
• Height, depth and thickness of each beam section.
The first two bullet points were decided to be kept still. The first one because of architectural consideration. The second one was set so that the angle between two crossing beams would be as near as possible to 90°. This way all the glass panels have four 90° degrees angles, which is the best shape possible because of the well known angle problems related to glass panels. The last bullet point is what is really relevant to us. Cross sections are what really drive structural failure and global stiffness requirements.
Output The output parameters are a lot actually. In fact, the global output is the entire
structure, not only by the architectural point of view, but also its mechanical behavior. In particular, in this first step, the beam stress will be the output taken into consideration.
Goal function Inputs affect outputs by means of the grasshopper structure definition. The
goal function collects all the necessary inputs in a single function which must be minimized. The chosen parameter to be minimized is set to be the maximum stress in each beam section, using the Von-Mises equivalent stress
And in this specific case it would become
Where
• N is the normal load, • A is the beam section area, • M is the bending moment, • W is the plastic section modulus, • Mt is the twisting moment,
• Ω is the area inside the middle line of the beam section, • s is the beam section thickness.
7.9.1 Goal: beam utilization factor with non-linear
weighted-average
Once a chromosome is set, karamba runs the analysis and returns the utilization factor UF for each beam. The value is set as a function v (UF) of this utilization factor, here I used the following expression:
With UF = k UF, where I set k=0.8 as a safety factor. This way, failure values of UF are being penalized. Then, the goal is
This algorithm took a lot of time with a meager result. This is comprehensible since the number of genes are huge (~2400).
8 Chapter
Evolutionary Multi-Criterion
Optimization
8.1 Introduction
Multi-Criterion Optimization (also known as multi-objective programming, vector optimization, multi-objective optimization, multi attribute optimization or Pareto optimization) is an area of multiple criteria decision making, that is concerned with mathematical optimization problems involving more than one objective function to be optimized simultaneously. Multi-objective optimization has been applied in many fields of science, including engineering, economics and logistics (see the section on applications for detailed examples) where optimal decisions need to be taken in the presence of trade-offs between two or more conflicting objectives. Minimizing weight while maximizing the strength of a particular component, and maximizing performance whilst minimizing fuel consumption and emission of pollutants of a vehicle are examples of multi-objective optimization problems involving two and three objectives, respectively. In practical problems, there can be more than three objectives. Multi-Criterion Optimization is really handy, if not the only feasible, when dealing with more conflicting objectives. When there are more than one objective to minimize (if one objective v must be maximized, one simply may minimize _v) there are two options possible:
• Creating a value function, which weight all inputs. This function is likely to be non-linear and most importantly, either the dependencies of those value are known or they are arbitrarily;
• Don’t create any value function, and retrieve the Pareto Front of the problem, set of optimal solutions. Without additional subjective preference information, all Pareto optimal solutions are considered equally good (as vectors cannot be ordered completely).
A solution is called non-dominated, Pareto optimal, Pareto efficient or non-inferior, if none of the objective functions can be improved in value without degrading some of the other objective values. Researchers study multi-objective optimization problems from different viewpoints and, thus, there exist different solution philosophies and goals when setting and solving them. The goal may be to find a representative set of Pareto optimal solutions, and/or quantify the trade-offs in satisfying the different objectives, and/or finding a single solution that satisfies the subjective preferences of a human decision maker (DM).
EMO, short name for Evolutionary Multi-Criterion Optimization, uses Genetic Algorithms to find non-dominated solutions. This way, each generation approaches more and more the zero vector 0 (see figure 9.1.1).
Figure 8-a Evolution of the Pareto front
Generating the Pareto 9.3 set can be computationally expensive and is often infeasible, because the complexity of the underlying application prevents exact methods from being applicable. For this reason, a number of stochastic search strategies such as evolutionary algorithms, tabu search, simulated annealing, and ant colony optimization have been developed: they usually do not guarantee to identify optimal trade-offs but try to find a good approximation, i.e., a set of solutions whose objective vectors are (hopefully) not too far away from the optimal objective vectors. Roughly speaking, a general stochastic search algorithm consists of three parts:
• a working memory that contains the currently considered solution candidates; • a selection module;
Figure 8-b Components of a general stochastic search algorithm
As to the selection, one can distinguish between mating and environmental selection. Mating selection aims at picking promising solutions for variation and usually is performed in a randomized fashion. In contrast, environmental selection determines which of the previously stored solutions and the newly created ones are kept in the internal memory. The variation module takes a set of solutions and systematically or randomly modifies these solutions in order to generate potentially better solutions. In summary, one iteration of a stochastic optimizer includes the consecutive steps mating selection, variation, and environmental selection; this cycle may be repeated until a certain stopping criterion is fulfilled.
